Solving for k,κ, and λgives
k= r2m
~2 (E±µB), (5.19)
and
κ= rm
~2 q
E±µB−V −p
−4U2+ (E±µB−V)2, (5.20) and
λ=
r8mU
~2κ2. (5.21)
Depth (Å)
%
80
00 160 240 320 400
20 40 60 80
O
Zr
C
Depth (Å)
%
620
00 1280 1920 2560 3200 20
40 60
80 Zr
O C
Figure 5.2. Depth profile of Zr foils cleaned with Citranox for both 5 ˚A and 50 ˚A resolution.
Courtesy: Russell Mammei.
UCN to enter.
5.2.3 Results
We measured the transmission through zirconium foils, as well as no foils, and fit an inter- polating function to the results. We can analyze the longitudinal spectrum by fixing the transmission at zero magnetic field to 100% transmission. Next an interpolating function was fit to the no-foil curve. In theory, this curve could be used to fit the reflection function of the various curves, but we believe that in practice this would require a Monte Carlo of the source-generated spectrum in order to separate the longitudinal spectrum from the full omnidirectional spectrum [72]. The initial raw data collected is shown in figure 5.4 as a function of longitudinal energy across the foil. For the case when there is no foil present in the PPM, the transmission decreases as the PPM magnetic field is increased. This is due to the splitting between polarization states. Low-field seekers are unable to penetrate the magnetic barrier, so only half of the UCN are able to pass through the field. When there is a foil, the transmission actually increases with the field. This is because the Fermi potential
2
3 4 6
5
1
4 7
9 10
8
11
Figure 5.3. Experimental setup for the foil transmission measurements; (1) UCN; (2) Gate valve monitor port; (3) UCN guide; (4)3He UCN detector; (5) Gate valve; (6) Valve door;
(7) Foil; (8) PPM; (9) PPM current coils; (10) PPM yoke; (11) Elephant trunk.
of the foil (90 neV) is enough to prevent a large fraction of UCN from penetrating. As the field is increased, half the neutrons (the high-field seekers) are boosted past this potential, but the fraction of the spectrum that is able to penetrate the foil barrier is still greater than that at zero magnetic field. By dividing the transmission through each foil thickness by the transmission with no foil, we obtain the relative transmission fraction as a function of longitudinal energy as shown in figure 5.5 The foil transmission,|T|2, can be be used to
d (µm) 0 T 1 T 2 T 3 T 4 T 5 T 6 T
0 neV 60 neV 120 neV 180 neV 240 neV 300 neV 360 neV
25.4 0.155 0.318 0.593 0.788 0.859 0.877 0.882
50.8 0.153 0.332 0.605 0.781 0.841 0.857 0.861
101.6 0.109 0.265 0.522 0.702 0.767 0.784 0.788
Table 5.1. Foil transmission as a function of Zr foil thickness and magnetic field strength and longitudinal energy. Longitudinal energy is listed below the magnetic field values.
extract the surface and bulk effects via
|T|2(x, E`) =c(E`) +ξ(E`)x, (5.22) by fitting the constants c(E`), the surface term, and ξ(E`), the bulk term, where x is the foil thickness. The results are presented in table 5.2. Some of this surface and bulk loss
Figure 5.4. A family of plots of absolute transmission of UCN through Zr foils as a function of thickness.
B 0 T 1 T 2 T 3 T 4 T 5 T 6 T
E` 0 neV 60 neV 120 neV 180 neV 240 neV 300 neV 360 neV
c(E`) 0.177 0.352 0.635 0.828 0.896 0.914 0.918
ξ(E`) 0.000635 0.000794 0.00103 0.00118 0.00123 0.00125 0.00125 Table 5.2. Foil transmission coefficients as a function of magnetic field strength.
is likely surface dependent [73–75]. We find that the constant component of the relative transmission,c(E`), increases dramatically with longitudinal energy. This is to be expected as the quantum mechanical reflection off the surface will decrease with increased energy.
The bulk transmission per distance,ξ(E`), also increases, as we would expect from the 1/v dependence of the cross section of the bulk material.
5.2.4 Summary
We measured the transmission of UCN through a zirconium foil compared to the transmis- sion with no foil. This foils were place inside of a polarizing magnet and the field was varied so as to alter the longitudinal spectrum of UCN passing through the foils. Bulk loss was measured separately from surface loss by measuring varying thicknesses of foils. We derived the surface loss from a thin layer, represented by a delta function, on the surface of a foil.
What this told us is that there is inherent loss on the surface of the Zr foil that is not in the
Figure 5.5. A family of plots of relative transmission of UCN through Zr foils as a function of thickness. The lines are arc tangent extrapolations to the data.
bulk. We may be able to use this knowledge to design better foils with a treated surface that is designed to scatter and reflect less. Using the QM model and a good understanding of the real and complex components of the Fermi potential of the surface materials, we may be able to design foils with near zero surface loss. Since this is the dominant loss factor, we might expect foils with very high transmission factors well over 90%. This measurement allowed us to find a replacement for the aluminum foils which had less than 80% transmis- sion [76]. This improvement increased the UCN density available to the UCNA and other experiments downstream of the LANL UCN source.
Chapter 6
Cosmological Limits on Fierz Interference
Images of broken light, which dance before me like a million eyes, they call me on and on across the universe.
John Lennon
One important limit on the Fierz term in neutron beta decay can come from Big Bang nucleosynthesis (BBN). The Fierz term, bn, modifies the Standard Model neutron decay rate, ΓSM,
Γ(n→p+e−+ ¯ν) =
1 +bnme Ee
ΓSM(n→p+e−+ ¯ν). (6.1) In the primordial universe during the nucleosynthesis era, up to about 3 minutes after the Big Bang, neutrons and protons were in equilibrium. As the universe cooled, this equilibrium began to freeze out, and the n/p ratio began to rapidly lower as neutrons decayed or were bound up in light nuclei. Prior to and during this transition, when a positron is absorbed by a neutron, (instead of an electron being emitted as with decay) the bn term in the reaction rate will have an opposite sign from the decay rate equation [77],
Γ(n+e+→p+ ¯ν) =
1−bnme
Ee
ΓSM(n+e+→p+ ¯ν). (6.2) These rate altering effects can modify the n/pratio which then completely determines the
4He primordial abundance. [78] Thus we can turn this argument around and can deduce a limit onbnfrom the4He primordial mass fraction,Ypfamiliar from observational cosmology [29]. Yp can be determined today from observation of nebulae where less stellar formation
has occurred since the dawn of the universe, leaving the primordial gas fractions relatively pure. In these regions, the mass fraction of hydrogen,X, and helium,Y, are expected to be in ratios largely untouched by contaminants generated by supernova remnant gases. These regions of gas can tell us how the final state of the BBN nuclear reaction network ended a few minutes after the Big Bang.
The neutron lifetime also strongly affects then/p ratio. The recent lower values for the neutron lifetime τn [79], will also alter the predicted primordial helium abundance [28, 80].
We will use the most recent value from [17].